A New Generating Function of (q-) Bernstein Type Polynomials and their Interpolation Function

Abstract

The main object of this paper is to construct a new generating function of the (q-) Bernstein type polynomials. We establish elementary properties of this function. By using this generating function, we derive recurrence relation and derivative of the (q-) Bernstein type polynomials. We also give relations between the (q-) Bernstein type polynomials, Hermite polynomials, Bernoulli polynomials of higher-order and the second kind Stirling numbers. By applying Mellin transformation to this generating function, we define interpolation of the (q-) Bernstein type polynomials. Moreover, we give some applications and questions on approximations of (q-) Bernstein type polynomials, moments of some distributions in Statistics.

References in corpus (1)

• Iterated Bernstein polynomial approximations

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• Some identities for Bernoulli polynomials involving Chebyshev polynomials
• A note on the values of the weighted q-Bernstein polynomials and modified q-Genocchi numbers with weight alpha and beta via the p-adic q-integral on Zp
• Some classes of generating functions for generalized Hermite- and Chebyshev-type polynomials: Analysis of Euler's formula
• Application of a composition of generating functions for obtaining explicit formulas of polynomials
• On p-adic analogue of q-Bernstein polynomials and related integrals
• On the q-Euler numbers related to modified q-Bernstein polynomials
• Generating functions for the Bernstein polynomials: A unified approach to deriving identities for the Bernstein basis functions
• On q-Berstein and q-Hermite polynomials
• A note on q-Bernoulli numbers and q-Bernstein polynomials
• q-Bernstein polynomials, q-Stirling numbers and q-Bernoulli polynomials
• A new family of q-Bernstein polynomials: Probabilistic viewpoint